Midterm 1  Feb 21, 2014 Math 484  25 questions, 2 pts each.   Section 002.Version B

Name _________________________________

No electronics is allowed.   Write both answers and details.

1. 2x -  1    is

(A) a linear equation for x, y, (B) an affine function of x,y  (C) a linear form in x,

(D) none of the above.

2.   x + y ≥ 0  is

(A) a linear constraint for x, y,  (B) a linear equation for x in standard form,

(C) a linear form in  x, y,  (D) an equation which cannot be solved, (E) none of the above.

3. The linear program  2x+y -> max subject to x+y=1

(A) is unbounded,  (B) is infeasible,  (C)  has many optimal solutions,

(D) has optimal value 3,   (E) none of the above.

4. The optimal value for the linear program   ax + b-> min, 0 ≤ x ≤ 1  with an unknown  x and given numbers  a, b is

(A)  it is not a linear program , (B)   the  program unbounded,

(C) b or a + b,   (D) none of the above.

5. The mathematical program x/y ->max subject to |x| ≤ 1, y ≥ 1 for two unknowns x, y

(A) is a linear program, (B) is infeasible, (C) has an optimal solution,

(D) is unbounded, (E) is not a mathematical program.

6. The mathematical program  x -> max,  x ≤ 1

(A) is not a linear program, (B) is infeasible, (C) has an optimal solution,

(D) is unbounded, (E) none of the above.

7. The answer to the mathematical program  4x+3y -> min , x2+ y2    25 is

(A) max= 25  at x = 4,  y = 3,  (B) max = 30  at x = 4, y = 3,  (C) max = 50 at x = 5, y =5,

(D) min =-25 at x = -4, y = -3,  ( E) none of the above.

8. The minimal total time for the job assignment problem

3 2 1 4

3 0 2 1

2 5 3 6

3 1 0 0

is    (A) 4,  (B) 6,  (C) 7, (D) 8, (E) none of the above.

9. For each  number  t, the equation     cos(t)x= sin(t)  for unknown x

(A) is not linear, (B) has a solution, (C) is a linear equation, (D) has many solutions,  (E) none of the above.

10. 0 ≤ 0 provided that x = y.   A) True,  B) False

11.  x > 0 and  y > 0   if  xy <  0.   A) True,  B) False

12. x > 0 only if x  ≥ 2.   A) True, B) False

13. The linear program given by a  standard tableau with the matrix

 -1 -2 -1 -3 0 -4 0 -2 0 1 0 1 0 2 -3

(A) is infeasible, (B) is unbounded,  (C) has an optimal solution,

(D) has only 2 linear constraints, (E)  none of the above.

14.  The linear program given by the standard tableau

x1 x2 x3 x4 1

1 2 0  -3 0      =x5

4 0 -2 0 1        =x6

0    -1 0 2 -3 -> min

(A) is infeasible, (B) is unbounded., (C) has an optimal solution,

(D) requires at least one pivot step to solve by simplex method.

15. The linear program given by a  standard tableau with the matrix

 1 2 0 -3 0 -4 0 -2 0 1 0 -1 0 2 -3

(A) is infeasible, (B) is unbounded, (C) has an optimal solution,

(D) has -3 as the optimal value, (E) has only one feasible solution.

16. The standard tableau

x1 x2 x3  1

-1   0    0   0 -> min

(A) is optimal, (B) has a bad column, (C) has a bad row, (D) is not standard, (E) none of the above.

17. The standard tableau

1

-1 = x1

0  = x2

-1 -> min

(A) is optimal. (B) has a bad column, (C) has a bad row, (D) is not standard, (E) is too small.

18. Pivoting the tableau

x     -y

2*  3 = z

produces the tableau

A) z    -y               B)  x   y

1/2 -3/2 = x               2           -3/2 = z

C) x y             D) z y             E) x y

1/2 3 = z          1/2 3/2 =x          1/2 -3/2 = z

19. The linear program x+y -> max, x≤  -1, y < -2

A) is infeasible. B) is bounded. C) has an optimal solution.

D) has infinitely many optimal solutions. E) has  1 as the optimal value.

20.The equation  x3+ y3  = 0 for x, y

A) has no solutions,  B) has exactly one solution, C) has infinitely many solutions.

21.The system x+ 2y = 3, 4x + 8y = 1   for x,y

A) has no solutions. B) has exactly one solution. C) has infinitely many solutions.

22. The set of constraints x + y > 2, 2x + 3y > 4 implies that

A) 4x + 5y ≥  9,  B) x ≥ 0,   C) y ≥ 1,   D) x ≤ 0,   E) none of the above.

23. For any number  x,    0 = 1 implies that  x ≥ 2 and x ≤ 1.   A) True B) False

24.  For any number  x,  if    x2 = 0 then     x ≥ 0 or x ≤ 0.     A) True   B)False

25. For any column  x  of numbers,    if 0 = 0 then x ≥ 0 or -x ≥ 0.   A) True  B) False